Cone, Cylinder, Sphere: Pouring Proofs

Also called

volume of a cone · volume of a cylinder · volume of a sphere · volume of a hemisphere · curved surface area of a cylinder · curved surface area of a cone · slant height · sector of a circle · one third of the cylinder · two thirds of the cylinder · pouring experiment · πr²h · ⅓πr²h · ⅔πr³ · 2πrh · πrl · शंकु का आयतन · बेलन का आयतन · गोले का आयतन

Topics

• A cone and a cylinder with the same base and height: three cones of sand or water exactly fill the cylinder.
• So the volume of a cone is one third of πr²h, the volume of the cylinder.
• A sphere of radius r fits exactly in a cylinder of radius r and height 2r; poured in, it fills two thirds of it.
• So the volume of a sphere is ⅔ × πr² × 2r = 4⁄3 πr³.
• Each hemisphere fills one third of that cylinder: its volume is ⅔ πr³, the same as two cones of radius r and height r.
• The curved surface of a cylinder unrolls into a rectangle 2πr wide and h high, so its area is 2πrh.
• The curved surface of a cone unrolls into a sector of radius l (the slant height) whose arc is 2πr, so its area is πrl.
• The slant height, height and radius of a cone make a right triangle: l² = h² + r².

About this model

A GLB file of about 292 KB, shown in Lably's 3D viewer and, on a phone, placed in your room with AR. Credit: Model, science and animation: Lably (built procedurally with three.js; no downloaded meshes). | CC BY 4.0. Original geometry and animation made for Lably.

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